This book presents fundamental concepts and notions related to the area of ranks in groups. Class-tested worldwide by highly qualified authors in the fields of abstract algebra and group theory, this book focuses on critical concepts with the most interesting, striking, and central results. In order to provide readers with the most useful techniques related to the various different ranks in a group, the authors have carefully examined hundreds of current research articles on group theory authored by researchers around the world, providing an up-to-date, comprehensive treatment of the subject.

• All material has been thoroughly vetted and class-tested by well-known researchers who have worked in the area of rank conditions in groups

• Topical coverage reflects the most modern, up-to-date research on ranks of groups

• Features a unified point-of-view on the most important results in ranks obtained using various methods so as to illustrate the role those ranks play within group theory

• Focuses on the tools and methods concerning ranks necessary to achieve significant progress in the study and clarification of the structure of groups

Ranks of Groups: The Tools, Characteristics, and Restrictions is an excellent textbook for graduate courses in mathematics, featuring numerous exercises, whose solutions are provided. This book will be an indispensable resource for mathematicians and researchers specializing in group theory and abstract algebra.

MARTYN R. DIXON, PhD, is Professor in the Department of Mathematics at the University of Alabama.

LEONID A. KURDACHENKO, PhD, DrS, is Distinguished Professor and Chair of the Department of Algebra at the University of Dnepropetrovsk, Ukraine.

IGOR YA SUBBOTIN, PhD, is Professor in the Department of Mathematics and Natural Sciences at National University in Los Angeles, California.

Preface

Chapter 1. Essential Toolbox 1

1.1. Ascending and Descending Series in Groups 1

1.2. Generalized Soluble Groups 7

1.3. Chernikov Groups and the Minimum Condition 19

1.4. Linear Groups 25

1.5. Some Relationships Between the Factors of the Upper and Lower Central Series 31

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